SYMMETRY, STATISTICAL MECHANICS AND RANDOM MATRICES

The theory of random matrices appears in many parts of mathematics such as probability, statistics, quantum chaos, number theory and the spectral theory of random Schrödinger operators. This lecture will give a brief introduction to the history and conjectures of this subject. We show how certain models of statistical mechanics provide a dual representation for spectral problems in random matrix theory. This representation enables one to obtain numerous identities arising from symmetry and to apply new
tools of analysis and phase transitions. Ordered and disordered phases correspond to different spectral types and time evolution of a random matrix Hamiltonian. A particular statistical mechanics model is equivalent to a history dependent random walk which prefers to jump to vertices it has visited in the past. The phase transition for this process is reflected in a change of the long time behavior of the walk.